Geodesic
Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities I. On the continuability of smooth solutions
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 693-718

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A class of nonlinear parabolic systems with quadratic nonlinearities in the gradient (the case of two spatial variables) is considered. It is assumed that the elliptic operator of the system has a variational structure. The behavior of a smooth on a time interval $[0,T)$ solution to the Cauchy-Neumann problem is studied. For the situation when the ``local energies'' of the solution are uniformly bounded on $[0,T)$, smooth extendibility of the solution up to $t=T$ is proved. In the case when $[0,T)$ defines the maximal interval of the existence of a smooth solution, the singular set at the moment $t=T$ is described.
Classification : 35B60, 35D05, 35J65, 35K50, 35K55
Keywords: boundary value problem; nonlinear parabolic systems; solvability 
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     author = {Arkhipova, A.},
     title = {Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities  {I.} {On} the continuability of smooth solutions},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {693--718},
     publisher = {mathdoc},
     volume = {41},
     number = {4},
     year = {2000},
     mrnumber = {1800172},
     zbl = {1046.35047},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a4/}
}
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Arkhipova, A. Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities  I. On the continuability of smooth solutions. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 4, pp. 693-718. http://geodesic.mathdoc.fr/item/CMUC_2000__41_4_a4/